Mark W. Ellis, Ph.D., NBCT
The challenge of helping students make sense of modeling with mathematics begins with the word itself. Although in everyday language the word “model” refers to a small-scale replica of a real object, in mathematics a model is a representation of a real-world object or situation.
A familiar example might be weather forecasts, which are based on mathematical models that involve many elements to predict what might happen. For complex phenomena like weather, there is not one “correct” model but many different models. Different models with varying degrees of accuracy are constantly being refined and revised. In talking about modeling, Zalmin Usiskin explains that the match between the real world and a mathematical model can range from exact (as in a model of combining nine books with 15 books) to impressionistic (as in models for weather forecasts).
Modeling is especially challenging when students have only experienced math as abstract rules and procedures or as contrived problems to be solved with a recently learned algorithm. In order to become proficient, students must regularly engage in modeling that offers opportunities to apply their learning. They need to look at real-world situations and think about what elements might be represented mathematically. Then they can collect data, generate mathematical expressions, check the data against the original situation, and make revisions.
Modeling involves several steps that may or may not be sequential:
- Identify a problem situation
- Make a representation of one or more elements of the situation
- Create a mathematical expression
- Compare results or predictions from the mathematical model with the real situation
- Make revisions to the model if needed
“When given a problem in a contextual situation, mathematically proficient students at the elementary grades can identify the mathematical elements of a situation and create a mathematical model that shows those mathematical elements and relationships among them. The mathematical model might be represented in one or more of the following ways: numbers and symbols, geometric figures, pictures or physical objects used to abstract the mathematical elements of the situation, or a mathematical diagram such as a number line, a table, or a graph, or students might use more than one of these to help them interpret the situation.”
In order to become proficient, students must regularly engage in modeling that both supports and lets them apply their learning. Indeed, the authors of Adding It Up: Helping Children Learn Mathematics point out that modeling is an everyday activity: “Outside of school they [students] encounter situations in which part of the difficulty is to figure out exactly what the problem is. Then they need to formulate the problem so that they can use mathematics to solve it.”
For primary grade students, teachers can suggest strategies for representing simple situations. For example, Ready Mathematics Grade 4 Lesson 6 asks students to use bar diagrams to represent contextual problems requiring multiplication or division. Students create mathematical expressions and are asked to explain how each element of their diagram and expression connects to the original context, thereby reinforcing mathematical sense-making.
When challenged to create their own problems, students are able to exercise more creativity while deepening their understanding of math concepts, relationships, and skills.
The activities on the Thinking Blocks website help students learn to use rectangular block diagrams to represent and solve problems with content from addition of whole numbers to ratio and proportion. Mosa Mack Science Detective offers interdisciplinary explorations for grades 4–8 aligned to current science, math, and English standards.
And this example from NCTM’s Mathematics Teaching in the Middle School asks students to examine the issue of pelican population using visual and mathematical models, offering nice connections with science standards. Video Story Problems offer an example of modeling using technology tools in creative ways. Check out these fifth graders!
When working with modeling problems, always ask students to explain how their work—the visual model, mathematical model, and final answer—relates to the original problem context. This will strengthen the habit of looking for connections and checking that the methods and results make sense.
As students become more comfortable with modeling, encourage them to identify their own problems based on familiar situations that may not have an obvious solution and solve these problems using modeling. For a nice example, see the article, “Posing Problems that Matter: Investigating School Overcrowding.” When sixth-grade students at an urban middle school complained that their school was overcrowded, the teacher developed a unit on area, perimeter, and similar figures, and students were challenged to prove that their school was in fact too crowded and create a presentation to the school board.
This blog is the third in a series from Curriculum Associates about the most challenging math standards that complements they’re recently released white paper – “Mastering the Most Challenging Math Standards with Rigorous Instruction“.
For more see:
- Spotlight on Math: Strategies for Addressing the Most Challenging Math Standards
- Spotlight on Math: Strategies for Addressing the Most Challenging Math Standards – Measurement
- Spotlight on Math: Strategies for Addressing the Most Challenging Math Standards – Fractions
- Spotlight on Math: Strategies for Addressing the Most Challenging Math Standards – Statistics
Mark Ellis is a National Board Certified Teacher and professor of education at California State University at Fullerton. He is an author of Curriculum Associates’ Ready® Mathematics program.